Theorem satfrel 35561

Description: The value of the satisfaction predicate as function over wff codes at a natural number is a relation. (Contributed by AV, 12-Oct-2023.)

Assertion

Ref	Expression
satfrel	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))

Proof of Theorem satfrel

Dummy variables 𝑎 𝑖 𝑗 𝑢 𝑣 𝑥 𝑦 𝑧 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6834	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘∅))
2	1	releqd 5728	. . . . 5 ⊢ (𝑎 = ∅ → (Rel ((𝑀 Sat 𝐸)‘𝑎) ↔ Rel ((𝑀 Sat 𝐸)‘∅)))
3	2	imbi2d 340	. . . 4 ⊢ (𝑎 = ∅ → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑎)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘∅))))
4		fveq2 6834	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘𝑏))
5	4	releqd 5728	. . . . 5 ⊢ (𝑎 = 𝑏 → (Rel ((𝑀 Sat 𝐸)‘𝑎) ↔ Rel ((𝑀 Sat 𝐸)‘𝑏)))
6	5	imbi2d 340	. . . 4 ⊢ (𝑎 = 𝑏 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑎)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑏))))
7		fveq2 6834	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘suc 𝑏))
8	7	releqd 5728	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (Rel ((𝑀 Sat 𝐸)‘𝑎) ↔ Rel ((𝑀 Sat 𝐸)‘suc 𝑏)))
9	8	imbi2d 340	. . . 4 ⊢ (𝑎 = suc 𝑏 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑎)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))))
10		fveq2 6834	. . . . . 6 ⊢ (𝑎 = 𝑁 → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘𝑁))
11	10	releqd 5728	. . . . 5 ⊢ (𝑎 = 𝑁 → (Rel ((𝑀 Sat 𝐸)‘𝑎) ↔ Rel ((𝑀 Sat 𝐸)‘𝑁)))
12	11	imbi2d 340	. . . 4 ⊢ (𝑎 = 𝑁 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑎)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑁))))
13		relopabv 5770	. . . . 5 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}
14		eqid 2736	. . . . . . 7 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
15	14	satfv0 35552	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})
16	15	releqd 5728	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Rel ((𝑀 Sat 𝐸)‘∅) ↔ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}))
17	13, 16	mpbiri 258	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘∅))
18		pm2.27 42	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑏)) → Rel ((𝑀 Sat 𝐸)‘𝑏)))
19		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑏 ∈ ω) ∧ Rel ((𝑀 Sat 𝐸)‘𝑏)) → Rel ((𝑀 Sat 𝐸)‘𝑏))
20		relopabv 5770	. . . . . . . . . 10 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}
21		relun 5760	. . . . . . . . . 10 ⊢ (Rel (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}) ↔ (Rel ((𝑀 Sat 𝐸)‘𝑏) ∧ Rel {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
22	19, 20, 21	sylanblrc 590	. . . . . . . . 9 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑏 ∈ ω) ∧ Rel ((𝑀 Sat 𝐸)‘𝑏)) → Rel (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
23	14	satfvsuc 35555	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑏 ∈ ω) → ((𝑀 Sat 𝐸)‘suc 𝑏) = (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
24	23	ad4ant123 1173	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑏 ∈ ω) ∧ Rel ((𝑀 Sat 𝐸)‘𝑏)) → ((𝑀 Sat 𝐸)‘suc 𝑏) = (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
25	24	releqd 5728	. . . . . . . . 9 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑏 ∈ ω) ∧ Rel ((𝑀 Sat 𝐸)‘𝑏)) → (Rel ((𝑀 Sat 𝐸)‘suc 𝑏) ↔ Rel (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})))
26	22, 25	mpbird 257	. . . . . . . 8 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑏 ∈ ω) ∧ Rel ((𝑀 Sat 𝐸)‘𝑏)) → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))
27	26	exp31 419	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑏 ∈ ω → (Rel ((𝑀 Sat 𝐸)‘𝑏) → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))))
28	27	com23 86	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Rel ((𝑀 Sat 𝐸)‘𝑏) → (𝑏 ∈ ω → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))))
29	18, 28	syld 47	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑏)) → (𝑏 ∈ ω → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))))
30	29	com13 88	. . . 4 ⊢ (𝑏 ∈ ω → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑏)) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘suc 𝑏))))
31	3, 6, 9, 12, 17, 30	finds 7838	. . 3 ⊢ (𝑁 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Rel ((𝑀 Sat 𝐸)‘𝑁)))
32	31	com12 32	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑁 ∈ ω → Rel ((𝑀 Sat 𝐸)‘𝑁)))
33	32	3impia 1117	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  {crab 3399   ∖ cdif 3898   ∪ cun 3899   ∩ cin 3900  ∅c0 4285  {csn 4580  ⟨cop 4586   class class class wbr 5098  {copab 5160   ↾ cres 5626  Rel wrel 5629  suc csuc 6319  ‘cfv 6492  (class class class)co 7358  ωcom 7808  1^st c1st 7931  2^nd c2nd 7932   ↑_m cmap 8763  ∈_𝑔cgoe 35527  ⊼_𝑔cgna 35528  ∀_𝑔cgol 35529   Sat csat 35530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-goel 35534  df-sat 35537

This theorem is referenced by:  satfdmlem  35562  satffunlem1lem2  35597  satffunlem2lem2  35600